Influence of Schistosity Orientation on Failure Mode and Indirect Tensile Strength of Mica Schist

Abstract

The indirect tension test is an important laboratory test for rock characterization. The presence of rock fabric, such as schistosity, complicates the assessment of test results. One hundred and forty-five indirect tension tests were conducted on mica schist specimens to investigate the effect of schistosity orientation on failure mode and tensile strength. Tensile strength results did not provide a clear relationship between schistosity orientation and tensile strength, so the failure patterns were investigated. A new naming scheme for failure modes was developed, incorporating fracture patterns observed in the specimen faces and edges. The Single Mode failure group specimens had only one failure pattern that appeared on both specimen faces, either axial failure (seventy-three specimens), schistosity failure (six specimens), or out-of-plane failure (seven specimens). The Mixed Mode failure group had thirty-two specimens that exhibited one failure pattern on one face and another on the other. The Hybrid Mode failure group had twenty-seven specimens with multiple failure patterns on both specimen faces. It was noted that Mixed Mode and Hybrid Mode specimens with components of axial failure had higher indirect tensile strengths than specimens without elements of axial failures. Statistical analyses of the tensile strength data using Levene’s Test for equal variances and two-sample t-tests showed no statistical difference between the Mixed Mode and Hybrid Mode failure groups. However, there was a statistical difference between the tensile strengths of the Single Mode axial failure specimens and the combined Mixed Mode and Hybrid Mode failure groups. These results clearly emphasize that indirect tensile strength should be assessed using schistosity orientation and failure mode.

1. Introduction

The tensile strength of rock is an important parameter across the spectrum of rock engineering and mechanics disciplines. Tensile strength is the anchor for the Hoek–Brown and Mohr–Coulomb failure envelopes in stress space [1]. These failure criteria are used to design infrastructure in and supported by rock. Tensile strength is an important parameter for assessing and designing the stability of infrastructure, including subsurface openings [2,3], slopes [4], building stones [5], and borehole stability [4,6]. In rock mechanics, the tensile strength is related to the crack initiation limit [7] and the damage and spalling limit [8].

Laboratory-based methods to measure rock tensile strength include direct (uniaxial tensile) and indirect (Brazilian, ring, hoop, and bending) tests. Direct tensile strength measurements are straightforward but very difficult, mainly because of the potential eccentricity of the testing machine, axial loading, and machine–rock sample connection issues [1,4,6]. The observed scattering of the results can be problematic [9]. The most common method to determine the tensile strength of rock is the indirect tension test.

Two researchers independently developed the indirect tension test (also known as the Brazilian, diametrical compression, or split tension test) [10,11]. It is the most common indirect test because of the ease of specimen preparation, a straightforward test procedure, and the short duration of the tests [1,4]. Both ASTM and the International Society for Rock Mechanics have established test procedures for indirect testing procedures.

In this test, a disc-shaped specimen is placed on its side, and a compressive force is applied. The specimen fails due to tensile splitting into two halves when loaded along two symmetric peripheral arcs such that the stress field is not uniaxial. Crack initiation should commence in the center of the disc where the specific strain energy is considered a minimum [12] and not at the zone under that loading platen, according to the Griffith fracture criterion. The tensile fracture should be along the center of the rock sample via tensile cracks starting near the center of the specimen. Some researchers have perceived this as a criterion to invalidate the test result, and it is a significant question concerning the reliability of the indirect test to determine the tensile strength. The history, perceptions, and in-depth analyses of the indirect tensile test have been presented in the literature [1,4,6,13].

Complications arise when characterizing the indirect tensile strength of anisotropic rock because the tensile strength is a function of anisotropy. It is well established that all rocks exhibit some degree of anisotropy [14]. This anisotropy may be due to mechanical defects [15] and the rock mass’s genesis. The formation of rock masses typically results in transverse isotropy, where properties are similar in two directions but differ in the third direction. Transverse isotropy is found in all rock types, including intact foliated metamorphic rocks (slates and schists) and intact laminated, stratified, or bedded sedimentary rocks (sandstone, shales, and limestones), and, at large scale, in volcanic rock formations (tuff and basalt) and sedimentary formations comprising alternating layers of different rock types [16].

This paper investigates the indirect tensile strength and failure modes of mica schist specimens obtained from core samples from vertical boreholes, not from oriented rock blocks cored in the laboratory. The paper reviews recent studies investigating schistose rock’s tensile strength and failure modes. Next, the experimental program is described. The results, especially the enhanced grouping and nomenclature used for describing specimen failure modes and the statistical analyses, are novel contributions to indirect tension testing of schistose rocks. Finally, the results from this study are discussed within the context of results from recent studies.

2. Previous Studies of Indirect Testing of Transversely Isotropic Rock

Studies on the indirect tensile behavior of transversely isotropic rock fall into two categories: the effect of discontinuity orientation on indirect tensile strength and the effect of layer orientation on fracture patterns. A brief history of research on the indirect tensile strength of transversely isotropic rock is presented. An in-depth presentation of recent studies on schist is provided.

2.1. Indirect Tensile Strength

Since the late 1950s, numerous studies have investigated the effect of discontinuity orientation on the tensile strength of various transversely isotropic rocks. Such studies have been conducted on coal [17,18,19], gneiss and granulite [19,20,21], marble [9,22], sandstone [16,23,24,25], shale [26,27], and slate [28]. These studies typically present the indirection tensile strength as a function of anisotropy angle. Typically, the results show the highest indirect tensile strengths when loaded parallel and perpendicular to the anisotropy and lower indirect tensile strengths when loaded with the anisotropy at an angle to loading. Pertinent to this study are investigations of schistose rocks, and recent studies are described below.

Oriented rock blocks were used to obtain twenty specimens of quartz mica schist for indirect tension testing. The specimen face schistosity was oriented perpendicular to loading (0°), 30° to loading, and parallel (90°) to loading, and average tensile strengths were 1.2 MPa, 2.4 MPa, and 3.4 MPa, respectively. Interestingly, the highest tensile strength occurred when the schistosity was oriented parallel to loading, and the lowest tensile strength occurred when the schistosity was oriented perpendicular to loading [20].

Twenty-one specimens of Yeoncheon schist (from Korea) were obtained from oriented rock blocks. Specimen face schistosity was oriented perpendicular to loading (0°) through parallel to loading (90°) in fifteen-degree increments. Three specimens were tested at each orientation. The highest average tensile strength (approximately 17.5 MPa) occurred when the schistosity was oriented perpendicular to the loading, and the lowest tensile strength (approximately 2.5 MPa) occurred when the schistosity was oriented parallel to the loading. There was considerable scatter in the tensile strength data at schistosity orientations of fifteen and thirty degrees [29]. These findings contradict those presented in the previous study [20].

Indirect tension tests were conducted on staurolite andalusite schist (from Iran) specimens obtained from oriented laboratory coring from rock blocks. The specimen face schistosity orientations were parallel to loading (0°) through perpendicular to loading (90°) in fifteen-degree increments. The highest strength was when the schistosity was perpendicular to loading (average of 5.1 MPa) and lowest when the schistosity was at an angle of 30° degrees to loading (3.3 MPa) [30].

Specimens of mylonitic schist and crenulated schist were obtained from oriented rock blocks. The mylonitic schist was cored in two directions perpendicular to the rock lineation. The crenulated schist was cored in two directions perpendicular to the crenulation cleavage. Specimens were tested with face schistosity orientations parallel to loading (0°) through perpendicular to loading (90°) in fifteen-degree increments. Two specimens were tested at each orientation. The average tensile strengths showed a variation as a function of schistosity orientation. The maximum tensile strength value for the mylonitic schist was perpendicular to loading (approximately 10 MPa). In contrast, the maximum tensile strength value for the crenulated schist was parallel to loading (about 6.8 MPa). The minimum tensile strength for the mylonitic schist occurred when the schistosity was orientated between 15 and 30 degrees, and the minimum tensile strength for the crenulated schist occurred when the schistosity was oriented between 45 and 60 degrees [31].

2.2. Fracture Patterns

Fracture pattern or failure mode studies often, but not always, accompany studies on the influence of discontinuity orientation on indirect tensile strength. This indicates the importance of fracture pattern assessment. One of the seminal studies of fracture patterns from indirect tensile tests on a transversely isotropic rock was conducted on sandstone specimens obtained from oriented rock blocks. Three distinct fracture patterns were identified: fractures parallel to isotropic layers (termed layer activation), fractures roughly parallel to loading that occur within 10% of the diameter on either side of the central line (termed central fractures), and fractures outside the central portion and do not correspond to layer activation fractures (termed non-central fractures) [24].

Fracture patterns were assessed in the study investigating quartz mica schist [20]. For specimens loaded parallel to the schistosity, the primary fracture was continuous and passed through the center of the specimens. However, two specimens showed fractures inclined to the loading direction. For specimens loaded perpendicular to the schistosity, the fractures were less continuous and appeared to have an en-echelon appearance.

For the Yeoncheon schist specimens [29], the fractures passed through the center of the specimen for discontinuity orientations between 0 (perpendicular) and 15 degrees. Specimens failed along the schistosity at orientations greater than 30 degrees. At schistosity orientations greater than 75 degrees, the failure plane again passed through the center of the specimen.

A study that tested an apatite–magnetite–tourmaline–biotite–chlorite bearing quartz-rich schist did not report indirect tensile strength as a function of schistosity orientation but assessed failure modes as a function of tensile strength. The cores from which the specimens were obtained were collected from a mine in Jharkhand, India. The schistosity orientation on the core axis varied from 10 to 40 degrees. They generally followed the naming scheme developed in previous studies [24]. The four failure modes for the schist were central (a single central fracture), non-central (failure not through the central portion but not following discontinuities), central+layer activation, and non-central+layer activation. Most schist specimens failed under a combination of two failure types, either central+layer activation or non-central+layer activation [32].

For the mylonitic schist and crenulated schist specimens [31], three fracture patterns were identified: layer activation, central fracture, and non-central fracture following previous nomenclature [24]. For the mylonitic schist, specimens with schistosity orientations of 0° (parallel) and 90° (perpendicular) to loading exhibited central fractures, whereas all other specimens exhibited mixed-mode layer activation and non-central fractures. Between discontinuity orientations of 0° to 60°, layer activation increased. This trend reversed from 60° to 90°. The crenulated schist fracture patterns were similar to the mylonitic schist fracture patterns. Between schistosity orientations of 15° to 45°, layer activation increased. This trend reversed from 45° to 90°. However, the mixed-mode fractures had an en-echelon appearance similar to those identified in a previous study [20].

3. Experimental Program

3.1. Core Sampling

Rock core samples were collected as part of a geotechnical characterization project for a deep foundation. The project location was near Flowery Branch, Hall County, Georgia, United States, located approximately 75 km northeast of Atlanta, GA, as shown in Figure 1. Geologically, the project area is within the Brevard Belt [33] or the Brevard Fault Zone [34]. The bedrock is described as a metagraywacke/mica schist of Precambrian to Paleozoic age [35].

Vertical geotechnical borings, consisting of standard penetration tests through soil and rock coring, were conducted to characterize the subsurface conditions along and next to a small railway embankment near Lake Lanier. Standard penetration tests (SPTs) were conducted until refusal at the top of the bedrock. Casing was set through the overburden soil to prevent the hole from collapsing. The bedrock was then cored with an NQ-sized diamond-studded bit fastened to the end of a hollow double-tube core barrel using standard 1.52 m (5-foot) runs. Cuttings were brought to the surface during coring by circulating water. Rock core samples from the bedrock were protected and retained in a swivel-mounted inner tube. Upon completion of each core run, the core barrel was brought to the surface, and the core samples were removed and placed in boxes. Core samples were logged in the field and then transported to the laboratory, where the samples were described, and core recovery and rock quality designation (RQD) were calculated.

Eight borings were conducted as part of the site characterization activities. The locations of the eight borings are shown in Figure 2. Five borings were performed near a north–south-oriented culvert at railway milepost 594.80, and three borings were performed near a north–south-oriented culvert at railway milepost 594.85. The eight borings were closely spaced, given the culverts were 0.05 miles or 80.5 m apart.

Rock coring, using an NQ-sized bit, was performed in three of the borings: B-2, B-2A, and B-3A. Rock coring commenced at depths of auger refusal in partially weathered rock. The thickness of the partially weathered rock ranged between 0.46 and 0.91 m. The length of the core obtained in the three boreholes was 7.6, 8.5, and 7.6 m from B-2, B-2A, and B-3A. Recoveries ranged between 70 and 100 percent within each run, with the lowest recoveries within the first run after the partially weathered rock. Rock quality designation ranged between 52 and 100 percent, with the lowest RQD values within the first run after the partially weathered rock.

3.2. Specimens

The core samples were brought to the first author’s laboratory at the University of North Florida in Jacksonville, Florida. Disc-shaped specimens were cut from the rock cores using a wet diamond saw to produce specimens with a thickness-to-diameter ratio of 0.2 to 0.75, per ASTM and ISRM standards. After cutting, the specimens were washed and oven-dried. The specimens were then weighed and measured using an analog dial gauge caliper. Two orthogonal diameters and four orthogonal thickness measurements were made on each specimen. One hundred and forty-six indirect tension specimens were prepared and tested in this study.

Table 1. Specimen data.

Parameter	Maximum	Average	Minimum	Standard Deviation
Mass (g)	137.1	122.3	102.5	6.49
Thickness (mm)	28.5	25.1	20.8	1.33
Thickness to Diameter	0.60	0.53	0.44	0.03
Density (g/cm3)	3.0	2.7	2.4	0.05
Unit Weight (kN/m3)	29.6	26.9	23.9	0.55

contains the specimens’ measurements, weight, and density data.

The specimen masses ranged between 137.1 and 102.5 g, averaging 122.3 g. Specimen thicknesses ranged between 28.5 mm and 20.8 mm, averaging 25.1 mm. According to ASTM and ISRM standards, the thickness-to-diameter ratio must be between 0.2 and 0.75. All specimen thickness-to-diameter ratios fell within this range and averaged 0.53. Specimen densities ranged between 3.0 and 2.4 g per cubic centimeter, averaging 2.7 g per cubic centimeter. The differences in densities are related to variations in mineralogy between specimens. The unit weight of the schist specimens varied between 29.6 and 23.9 kilonewtons per cubic meter.

The majority of the previous studies investigating the effect of schistosity on the indirect tensile strength and failure modes of schist used rock blocks cored in the laboratory. This provides researchers with highly controlled specimens cored at specific angles to the schistosity. The specimens used in this study are from vertical core borings into a rock mass. As a result, the specimens used in this study have a variety of specimen face schistosity, orientations, and thicknesses of edge schistosity. Figure 3 shows the wide variety of face and edge schistosity observed in the specimens.

The unconfined compression specimen in Figure 3 is an example of the orientation of the schistosity to the core axis. The drilling logs indicated the schistosity was oriented at angles between twenty and twenty-five degrees from horizontal. The specimen face schistosity consisted of a wide variety of fabrics. In most cases, the orientation of the schistosity was apparent, but a small minority of face schistosity fabrics tended towards no fabric (massive). Edge schistosity exhibited thin and thick schistosity bands. The specimen edge schistosity bands typically intersected one face of the specimen. However, there were cases when the schistosity would intersect both faces or neither face of the specimen.

3.3. Indirect Tension Testing

Indirect tension testing was conducted using a hand-operated hydraulic load frame manufactured by GCTS. Figure 4 contains pictures of the load frame. A lever-operated hand pump moves the lower platen up. The upper platen is fixed on a cross member between two steel rods. The system has a capacity of 100 kN and can be used for indirect tension testing, point load testing, and unconfined compression testing of weak rock.

The disc-shaped specimens were placed on their edge on the lower platen to start the tests. The specimen was rotated so the face schistosity was orientated at a predetermined angle. For this study, loading parallel to face schistosity was termed 0°, and loading perpendicular to face schistosity was termed 90°. The orientation of the schistosity on the specimen edge could not be controlled because, as previously discussed, specimens were obtained from vertical cores taken from the in situ rock mass and not from oriented rock blocks as in previous studies. Figure 5 shows the face and edge schistosity orientations, terminology, and examples of the face and edge schistosity of two specimens used in the study.

After the specimen was rotated to the desired face schistosity orientation, the specimen was held in place, and the bottom platen moved up so that the specimen was in contact with the upper platen. This ensured the specimen would not rotate from its desired orientation. The pressure gauge readout was zeroed, and the test began by slowly pumping the hand pump to compress the specimen. As required by ASTM procedures, specimen failure occurred within one to ten minutes of the start of loading. The maximum load (kN) to cause failure was recorded.

The tensile strength of the specimens was calculated using:

$${\sigma }_t=\frac{2P}{\pi LD} \left(\mathrm{M}\mathrm{P}\mathrm{a}\right)$$

(1)

where P is the maximum load applied to the specimen (N), L is the average thickness of the specimen (mm), and D is the average diameter of the specimen (mm).

After testing, photographs of each specimen’s front face, back face, top edge, and bottom edge were obtained. These photographs were used to classify the specimen failure modes.

4. Results

The results section presents the indirect tensile strength as a function of face schistosity orientation, failure mode classification, and indirect tensile as a function of failure mode and face schistosity orientation. Also included is a statistical analysis of tensile strengths associated with failure modes.

4.1. Indirect Tensile Strength as a Function of Schistosity Orientation

The relationship between indirect tensile strength and schistosity orientation is shown in Figure 6. There is considerable scatter in the data. The highest tensile strength is 15.1 MPa, and the lowest is 4.0 MPa. The largest difference in tensile strength for a given schistosity orientation is 10.24 MPa, which occurred at a schistosity orientation of approximately 25 degrees. The highest average tensile strengths occur between schistosity orientations of 50 to 60 degrees. The best-fit regression line was plotted, and was a second-degree polynomial with an R² value of 0.04. Since there was no clear relationship between tensile strength and orientation of schistosity, the relationship between tensile strength, schistosity orientation, and failure mode was investigated.

4.2. Specimen Failure Modes

The specimen failure classification is based on observed failure patterns on the specimen faces and edges but does not incorporate tensile strengths. The developed classification system consists of three groups: the Single Mode Failure Group, Mixed Mode Failure Group, and Hybrid Mode Failure Group. The group name indicates the number or complexity of fracture patterns within a specimen. The complexity of the fracture patterns increases from the Single Failure Group to the Hybrid Failure Group.

Different combinations of failure patterns are associated with each Group, as shown in

Table 2. Specimen failure mode classification.

Failure Mode Group (145 Specimens)
Single Mode Failure (86)	Mixed Mode Failure (32)	Hybrid Mode Failure (27)
Axial Failure (73) Schistosity Failure (6) Out-of-Plane Failure (7)	Axial & Schistosity (17) Axial & Out of Plane (7) Schistosity & Out of Plane (8)	Axial & Axial–Schistosity (16) Axial & Axial–Out of Plane (6) Out of Plane & Axial–Out of Plane (3) Axial–Schistosity & Axial–Schistosity (1) Axial & Schistosity–Out of Plane (1)

. The Single Mode Failure Group consists of specimens with a single failure pattern. The failure patterns within the Single Failure Mode Group are either axial failure, schistosity failure, or out-of-plane failure. The Mixed Mode Failure Group consists of specimens exhibiting one failure pattern on one face and another on the other. Specimens have either axial and schistosity failure, axial and out-of-plane failure, or schistosity and out-of-plane failure. The Hybrid Mode Failure Group has specimens with the most complex failure patterns. Specimens in this group have one or two failure patterns on one face and two failure patterns on the other face. The classification system is shown in Table 1. Each failure mode group, characteristics, and the number of specimens (in parentheses) are described below.

4.2.1. Single Mode Failure Group

Specimens within the Single Failure Mode Group exhibit a single failure characteristic on both specimen faces and edges. These failure characteristics are either axial failure, schistosity failure, or out-of-plane failure. Eighty-seven specimens are within the Single Failure Mode Group. Sketches and photographs of these failure characteristics are shown in Figure 7.

Axial failure occurs when the failure plane cuts across schistosity and is within the central 10 percent of the specimen. On the specimen edge, the failure plane also cuts across the foliation. This failure mode is considered the failure mode for a successful split tension test. It is common for the failure plane to be parallel to the compressive loading direction in homogeneous rock specimens [13]. Seventy-three specimens exhibited this failure mode. This failure mode occurred at all foliation orientations.

Schistosity failure occurs when the failure plane follows the schistosity and intersects the faces of the specimen. It is important to note that no aspect of axial failure is associated with schistosity failure. On the specimen edge, the failure follows the schistosity which intersects the specimen face. Only six specimens exhibited this failure mode. The schistosity orientation for this failure mode ranged between zero and approximately twenty degrees. For this type of failure to occur, the foliation must begin or terminate near the application of the load on the specimen face. This type of failure is considered a shear-type failure because it follows the schistosity of the specimen.

Out-of-plane failure occurs when the failure plane does not intersect the faces of the specimen. With this failure mode, failure occurs along schistosity within the specimen and does not intersect the specimen face. On the edge of the specimen, the failure plane follows the schistosity, but the schistosity does not intersect the specimen face. Only seven specimens exhibited this failure mode. The schistosity orientation for this failure mode was predominantly high angles to loading, between forty-five and ninety degrees, except for one specimen with a foliation orientation of approximately seventeen degrees. This type of failure is a shear failure because it follows the schistosity of the specimen.

The identification of the schistosity and out-of-plane failure modes highlights the importance of the edge schistosity orientation. Both are shear-type failure modes, but the edge schistosity orientation dictates which failure mode occurs.

4.2.2. Mixed Mode Failure Group

The same three failure characteristics are present in the Mixed Failure Mode Group. However, these specimens exhibit one type of failure on one face and a different failure type of failure on the opposite face. The naming convention for this group is the failure characteristic on one face, an ampersand, and the failure characteristic on the other face. Three combinations of mixed failure mode specimens were identified: Axial & Schistosity Failure, Axial & Out-Of-Plane Failure, and Schistosity & Out-Of-Plane Failure. There is no importance to the order of the failure names. Twenty-seven specimens fall into this group.

Axial & Schistosity Failure was seen in seventeen specimens. This type of failure occurred in specimens with schistosity orientations between zero and approximately seventy degrees. Schistosity & Out-Of-Plane Failure was seen in eight specimens. This type of failure had the broadest range of orientations between zero and about seventy-five degrees. Axial & Out-Of-Plane failure occurred in seven specimens. This type of failure had the smallest range of orientations: approximately fifty-five to ninety degrees.

4.2.3. Hybrid Mode Failure Group

The same three failure characteristics are also in the Hybrid Failure Mode Group. The Hybrid Failure Mode Group has the most complicated failures. The specimens in this group exhibit two failure characteristics on one face and one or two failure characteristics on the opposite face. The two failure characteristics may differ from those on the opposite face. There are thirty-one specimens in the Hybrid Failure Mode Group.

The Hybrid Failure modes are described using an ampersand separating the failure characteristics on each face and a dash between failure characteristics on the same face of the specimen. The order of the failure modes indicated in the group is arbitrary. There are five groups in the Hybrid Failure Mode Group:

• Axial & Axial–Schistosity: sixteen specimens with foliation orientations between zero and ninety degrees;
• Axial & Axial–Out of Plane: six specimens with foliation orientations between approximately forty and ninety degrees;
• Out of Plane & Axial–Out of Plane: three specimens with foliation orientations between approximately forty-five and fifty-five degrees;
• Axial–Schistosity & Axial–Schistosity: one specimen with foliation orientation of ninety degrees; and
• Axial & Schistosity–Out of Plane: one specimen with foliation orientation of approximately sixty-five degrees.

Figure 8 presents the schistosity orientations associated with the identified failure modes. The two failure modes represented by a circle indicate that only one specimen had that failure mode. In general, failure modes related to schistosity-type failure occurred at low schistosity orientations, whereas those associated with out-of-plane-type failure occurred at high schistosity orientations. This is seen in the Single Mode Failure Group. As the specimen failure becomes more complex, this relationship becomes somewhat blurred. One specimen exhibited out-of-plane failure at a low schistosity angle, which is considered an anomaly.

Within the Mixed Mode Failure Group, there is considerable overlap within the axial & schistosity and schistosity & out-of-plane failure types. However, like the Single Mode Failure Group, the axial & out-of-plane failures only occurred at high schistosity angles. Interestingly, the schistosity and out-of-plane failures did not occur at all schistosity orientations. This may be due to the limited number of specimens which exhibited this type of failure.

In the Hybrid Mode Failure Group, the axial & axial-schistosity failures occurred at schistosity orientations from zero to ninety degrees. The axial & axial-out of plane failures occurred at schistosity orientations between approximately forty and ninety degrees. This type of failure was expected to occur at all orientations, possibly due to the small number of specimens that failed this way. The other failure characteristics have so few specimens that meaningful conclusions could not be made.

4.3. Tensile Strength—Failure Mode Relationships

With the development of failure modes, assessing the tensile strength as a function of failure mode is now possible. These relationships will hopefully show how failure modes influence strength values. Additionally, such a relationship can provide insights into the schistosity orientations that should be tested to provide appropriate strength values.

4.3.1. Single Mode Failure Group Tensile Strengths

Figure 9 shows the relationship between schistosity orientation and tensile strength for the Single Mode Failure Group specimens. Specimens from the three distinct failure modes are highlighted in the figure. There is a clear distinction of tensile strength between failure modes. The specimens with axial failure mode generally had the highest tensile strengths. The average tensile strength of these specimens was 11.25 MPa with a standard deviation of 1.27 MPa.

The average tensile strengths for schistosity and out-of-plane failures were much lower. The average tensile strengths and associated standard deviations, in parentheses, are 7.37 MPa (0.95 MPa) and 7.67 MPa (2.14 MPa) for schistosity and out-of-plane failures, respectively. It is essential to consider that a limited number of specimens exhibited schistosity and out-of-plane failures.

The best-fit line for the Single Mode Failure Group specimens is a second-degree polynomial. When plotting only the Single Mode Failure Group specimens, there was a slight improvement to the R² value.

4.3.2. Mixed Mode Failure Group Tensile Strengths

Figure 10 shows the relationship between tensile strength and schistosity orientation for the Mixed Mode Failure Group specimens. Specimens from the three distinct failure modes are highlighted in the figure. Unlike the Single Failure Mode specimens, there is very little difference in tensile strength based on failure mode. The best-fit line is a second-degree polynomial with an R² value of 0.03. The average tensile strength of the Mixed Mode specimens is 9.2 MPa with a standard deviation of 1.59 MPa. The tensile strength and standard deviation are less than the combined specimens from the Single Mode Failure Group. This is expected, given these specimens exhibit failure modes that produce tensile lower strengths.

As expected, the average tensile strength is higher when the failure mode contains an axial failure. For the Mixed Mode Failure Group specimens, the axial & schistosity failures have the highest average tensile strength (9.5 MPa), followed by the axial and out-of-plane failures (9.01 MPa) and the schistosity & out-of-plane failures (8.7 MPa).

4.3.3. Hybrid Mode Failure Group Tensile Strengths

Figure 11 shows the relationship between tensile strength and foliation orientation for the Hybrid Mode Failure Group specimens. Specimens from the five distinct failure modes are highlighted in the figure. Once again, unlike the Single Mode Failure Group specimens, tensile strength does not differ based on failure mode. The average tensile strength of the Hybrid Mode Failure specimens is 10.05 MPa with a standard deviation of 1.97 MPa. This tensile strength is greater than the average Mixed Mode Failure Group specimens’ tensile strength but less than the Axial Failure tensile strength. The best-fit line through the data was once again a second-degree polynomial. The R2 value is 0.003.

4.4. Statistical Analysis of Tensile Strength Based on Failure Modes

Plotting the tensile strength as a function of schistosity orientation, as shown in Figure 6, demonstrated no visual relationship. It has been suggested that a post-test examination of specimens should be conducted to establish the validity of each test [36]. Valid indirect tensile tests show fracture initiation at the center of the specimen, fracture through the middle of the specimen, and no visible effects from the loading platens [18,19,36]. Using the fracture through the central portion of the specimen criteria, only Single Mode axial failure specimens constitute valid tests. This accounts for approximately fifty percent of the specimens. However, many of the Mixed Mode and Hybrid Mode failures had a component of axial failures. Is it appropriate to count these specimens as valid tests even though the specimen failure was not a fully axial failure?

Table 3. Summary of average tensile strengths and standard deviations.

Failure Mode Group		Number of Specimens	Tensile Strength (MPa)	Standard Deviation (MPa)
Single Mode	Axial	73	11.27	1.27
	Schistosity	6	7.37	0.95
	Out of Plane	7	7.66	2.13
Mixed Mode		32	9.17	1.59
Hybrid Mode		27	10.04	1.96

summarizes the average tensile strengths and their standard deviations from specimens tested in this study. The individual failure modes from the Single Failure Mode specimens are presented separately in the table because they are genuinely distinct failure modes. The Mixed Mode Failure Group, consisting of one failure mode on one face and a different failure mode on the other face, was grouped because of the limited number of specimens in some failure mode categories (as shown in Table 2 and Figure 8). The Hybrid Mode Failure Group was grouped because of the limited number of specimens in some of the failure mode categories, and every failure mode had a component of axial splitting (as shown in Table 2 and Figure 8).

In the Single Failure Mode specimens, the schistosity and out-of-plane specimens have tensile strengths lower than the axial splitting specimens. This is unsurprising because the schistosity and out-of-plane failures occur along the schistosity, which is the plane of weakness of the specimen. The out-of-plane schistosity specimens have higher average tensile strengths because the failure planes are longer than those of the schistosity failure mode, as shown in Figure 7.

The Mixed Mode average tensile strength and Hybrid Mode average tensile strength are similar and slightly below the Single Mode axial failure average tensile strength. Although the Mixed Mode and Hybrid Mode failure mode specimens are not considered valid tests because their failure partially follows schistosity and is not within the middle of the specimen, their tensile strengths may be statistically similar to the Single Mode Axial failure average tensile strength.

A suite of statistical tests was conducted on the tensile strengths and standard deviations of the Single Mode Group axial specimens, Mixed Mode Group, and Hybrid Mode Group data to assess whether they were statistically different. The statistical tests were performed in Minitab 18.

First, Levene’s Test for equal variances was conducted. This test assesses whether the standard deviations are significantly different between data sets. Next, an unpaired or independent t-test was performed using the appropriate assumptions from Levene’s Test to determine if there was a statistically significant difference between the mean tensile strengths.

The Mixed Mode Failure Group and Hybrid Mode Failure Group tensile strengths are similar, so they were the first datasets evaluated. Levene’s Test was conducted at a confidence interval of 95%. The p-value for the test was 0.4953, which was greater than the alpha value of 0.050, meaning the differences between the variances of the Mixed Mode and Hybrid Mode specimens were not significantly different.

The independent t-test was run, assuming the standard deviations were not significantly different. The results of the independent t-test, performed at a 95% confidence interval, showed there was no statistical difference between the tensile strength between the Hybrid Mode failure group and the Mixed Mode failure group tensile strengths. The p-value of 0.0608 was greater than the alpha value of 0.5.

Since the Mixed Mode failure group and Hybrid Mode failure group tensile strengths were not statistically significantly different, their data was combined into a single large dataset. This combined dataset was assessed with the Single Mode axial failure specimen data. Again, Levene’s Test was first conducted. The p-value for the test was 0.0156, which was less than the alpha value of 0.050, meaning the tensile strength variances were large enough to be statistically significant. The independent t-test results, assuming that standard variances were significantly different, were performed at a 95% confidence interval. The results showed the average of the combined Mixed Mode and Hybrid Mode population is not considered equal to the average of the Single Mode axial failure population. The t-value was −6.299, with an associated p-value of 0.000.

Although having an axial failure component increases the indirect tensile strength in a Mixed Mode or Hybrid Mode specimen, statistical testing has shown the combined Mixed Mode and Hybrid Mode specimen tensile strengths are statistically different than the Single Mode axial failure specimens. The statistical difference between Single Mode Axial tensile strengths and combined Mixed Mode and Hybrid Mode tensile strengths confirms that specimen failure modes must be considered when assessing the indirect tensile strengths of schist specimens.

5. Discussion

The discussion section compares this study with recent studies that investigated indirect tensile strength as a function of schistosity orientation and possible sources of error in the study. Also included is a discussion of directions for future work.

5.1. Comparison with Previous Studies

The most crucial difference between this study and previous studies assessing the influence of schistosity on indirect tensile strength is how the specimens were obtained. The previous studies [20,29,30,31] obtained specimens from rock blocks cored in the laboratory at specific orientations. This has two implications that help constrain their results: tensile strengths of specimens from the same orientations should be similar because small-scale defects and weathering effects should be similar within a rock block, and the orientation of edge schistosity will be similar. These can be considered scientific studies where there is considerable effort to control experimental variables. As a result, the indirect tensile strength as a function of schistosity orientation showed the classic “U-shaped” relationship with the highest indirect tensile strengths associated with schistosity oriented perpendicular and parallel to loading and lower indirect tensile strengths at schistosity inclined to the direction of loading. The current study showed no visible relationship between tensile strength and schistosity orientation, which required an assessment of indirect tensile strengths as a function of failure mode.

Previous studies generally incorporated the fracture pattern nomenclature developed for sandstone [24]. Studies [31,32] incorporated a mixed-mode description of fractures that included a layer activation term associated with failure along schistosity. The failure modes described in these studies were relatively simple because the variability of the specimen edge schistosity was minimized. The technique used to obtain specimens in the current study, which had variable edge schistosity orientations, necessitated a more complicated failure mode naming scheme. As shown in Figure 3 and Figure 7, edge schistosity varied from slightly dipping to intersect the specimen face, causing a schistosity failure, to steeply dipping so that it did not intersect the specimen face, causing an out-of-plane failure. The out-of-plane type failure was not noted in any previous studies.

One notable exception in the previous studies [32] is that specimens were collected from a mine, not from oriented rock blocks cored in the laboratory. They did not report tensile strength as a function of schistosity orientation, most likely because, like this study, there was no clear trend in the data. They noted that most specimens exhibited multiple failure types. They did note that the orientation of schistosity on core samples varied from 10° to 40°, but they did not incorporate the edge schistosity orientation into their study.

This study has an important place in the engineering literature. Often, geotechnical coring for foundation systems is only conducted in a vertical direction. The low budgets for geotechnical site characterization [37] often preclude in-depth studies of intact rock properties. In such cases, the geotechnical design must rely on properties obtained from non-orientated core samples. This study provides an example of such an analysis. Furthermore, the results of the statistical testing confirm that valid indirect tension tests must exhibit axial failures.

5.2. Possible Sources of Error and Limitations

The specimen preparation and testing used in this study are very straightforward and do not contribute to any sources of error. This study’s primary source of error was assessing the schistosity orientation and grain size within specimens. Figure 12 shows six examples of the schistosity on the face of the specimens used in this study. Most specimens had well-developed schistosity, so orienting them to a predetermined angle was easy. Specimens with less-developed and poorly developed schistosity were challenging to orient, and their reported orientations may not be accurate. Two specimens, prepared but not used in this study, were massive, meaning they had no visible schistosity.

There are two limitations associated with this study. One of the limitations, which can be seen in Figure 12, is the relative range of grain sizes within the specimens. The tensile strengths were not assessed according to the grain sizes within the specimens. The second limitation is the orientation and spacing of the schistosity on the specimen edges. The direction of this schistosity varied such that it intersected the specimen face or did not intercept the specimen face. The tensile strengths were not assessed with respect to the orientation of the schistosity on the edge of the specimen.

5.3. Directions for Future Work

Valid indirect tensile tests initiate fracture in the specimen’s central portion [36]. A post-test assessment of fracture modes or patterns provides no insight into whether a valid test occurred. Three methods can be employed to determine the location of fracture initiation: the use of high-speed cameras to visually detect the formation of microcracks [38], the use of acoustic emission monitoring to calculate the location of microcracking events [39], and the use of digital image correlations (DIC) to detect strain localization on the surface of the specimen [40]. Coupling these techniques to provide a full assessment of fracture initiation is possible. The expense of the equipment means these characterization activities must be conducted in research facilities rather than commercial laboratories.

One unexplored avenue of research is incorporating specimen face and specimen edge schistosity in assessing tensile strength. This type of investigation would be logistically tricky but not impossible because numerous rock blocks and specimens must be obtained and tested.

6. Conclusions

Indirect tension testing was performed on one hundred and forty-five schistose rock specimens obtained from vertical core borings. For the indirect tension testing, specimen face schistosity varied between zero degrees (parallel to the loading direction) to ninety degrees (perpendicular to the loading direction).

When the tensile strength was plotted as a function of schistosity orientation, there was considerable scatter to the data. The typical “U-shaped” relationship with high tensile strengths when specimens are loaded parallel and perpendicular to schistosity and lower strength when schistosity is at an angle to the loading direction was not observed.

The failure modes were assessed to help identify trends in the data, and a new nomenclature was developed to describe the failure modes. Three failure mode groups were identified: Single Mode Failure, Mixed Mode Failure, and Hybrid Mode Failure. The Single Mode Failure Group specimens exhibited either axial failure (failure across schistosity), schistosity failure (failure along schistosity), or out-of-plane failure (failure along schistosity that does not intersect the specimen face). The Mixed Mode Failure Group exhibited one single failure mode on one face and a different failure mode on the other. The Hybrid Mode Failure Group specimen exhibited two failure characteristics on one face and one or two on the opposite face. Failure modes correlated to the schistosity orientation on the face of the specimens. The failure mode nomenclature developed was similar to previous studies; however, this study included the schistosity orientation on the specimen’s edge in the failure descriptions. The assessment of failure modes demonstrated that Single Mode axial failure strengths were greater than Single Mode schistosity and our-of-plane failure strengths.

Average indirect tensile strengths from the Mixed Mode and Hybrid Mode Failure Groups were similar to the Single Mode axial failure specimens. Statistical analyses were performed to determine statistical similarities and differences within the data sets. Results showed no statistical differences between the tensile strengths of the Mixed Mode and Hybrid Mode Failure Group specimens. However, there was a statistical difference between the tensile strengths of the Single Mode axial failure specimens and the combined Mixed Mode and Hybrid Mode Failure Group specimens. These results demonstrate that it is essential to consider both schistosity orientation and failure modes when assessing the indirect tensile strengths of schistose specimens.